Results 1  10
of
12
Quantum diffusion of the random Schrödinger evolution in the scaling limit II. The recollision diagrams
"... We consider random Schrödinger equations on R d for d ≥ 3 with a homogeneous AndersonPoisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as x ∼ λ −2−κ/2,t ∼ λ −2−κ with 0 < κ < κ0(d). We prove that, in t ..."
Abstract

Cited by 37 (6 self)
 Add to MetaCart
We consider random Schrödinger equations on R d for d ≥ 3 with a homogeneous AndersonPoisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as x ∼ λ −2−κ/2,t ∼ λ −2−κ with 0 < κ < κ0(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψt converges weakly to the solution of a heat equation in the space variable x for arbitrary L 2 initial data. The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the nonrecollision graphs and prove that the amplitude of the nonladder diagrams is smaller than their “naive size ” by an extra λ c factor per non(anti)ladder vertex for some c> 0. This is the first rigorous result showing that the
Convergence of Perturbation Expansions in Fermionic Models. Part 2: Overlapping Loops
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2004
"... We improve on the abstract estimate obtained in Part 1 by assuming that there are constraints imposed by ‘overlapping momentum loops’. These constraints are active in a two dimensional, weakly coupled fermion gas with a strictly convex Fermi curve. The improved estimate is used in another paper to ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
We improve on the abstract estimate obtained in Part 1 by assuming that there are constraints imposed by ‘overlapping momentum loops’. These constraints are active in a two dimensional, weakly coupled fermion gas with a strictly convex Fermi curve. The improved estimate is used in another paper to control everything but the sum of all ladder contributions to the thermodynamic Green’s functions.
A Two Dimensional Fermi Liquid. Part 1: Overview
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2004
"... In a series of ten papers (see the flow chart at the end of §I), of which this is the first, we prove that the temperature zero renormalized perturbation expansions of a class of interacting many–fermion models in two space dimensions have nonzero radius of convergence. The models have “asymmetric ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
In a series of ten papers (see the flow chart at the end of §I), of which this is the first, we prove that the temperature zero renormalized perturbation expansions of a class of interacting many–fermion models in two space dimensions have nonzero radius of convergence. The models have “asymmetric ” Fermi surfaces and short range interactions. One consequence of the convergence of the perturbation expansions is the existence of a discontinuity in the particle number density at the Fermi surface. Here, we present a self contained formulation of our main results and give an overview of the methods used to prove them.
Determinant Bounds and the Matsubara UV Problem of Many–Fermion Systems
 Comm. Math. Phys
, 2008
"... Dedicated to Jürg Fröhlich in celebration of his 61 st birthday It is known that perturbation theory converges in fermionic field theory at weak coupling if the interaction and the covariance are summable and if certain determinants arising in the expansion can be bounded efficiently, e.g. if the co ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Dedicated to Jürg Fröhlich in celebration of his 61 st birthday It is known that perturbation theory converges in fermionic field theory at weak coupling if the interaction and the covariance are summable and if certain determinants arising in the expansion can be bounded efficiently, e.g. if the covariance admits a Gram representation with a finite Gram constant. The covariances of the standard many–fermion systems do not fall into this class due to the slow decay of the covariance at large Matsubara frequency, giving rise to a UV problem in the integration over degrees of freedom with Matsubara frequencies larger than some Ω (usually the first step in a multiscale analysis). We show that these covariances do not have Gram representations on any separable Hilbert space. We then prove a general bound for determinants associated to chronological products which is stronger than the usual Gram bound and which applies to the many–fermion case. This allows us to prove convergence of the first integration step in a rather easy way, for a short–range interaction which can be arbitrarily strong, provided Ω is chosen large enough. Moreover, we give – for the first time – nonperturbative bounds on all scales for the case of scale decompositions of the propagator which do not impose cutoffs on the Matsubara frequency. present address: Institut für Mathematik, Universität Mainz 1 1 Gram representations and determinant bounds Let X be a set and M: X 2 → C, (x, y) ↦ → M(x, y). We call M an (X×X)matrix and use the notation M = (Mxy)x,y∈X (if X = {1,..., n}, we call it as usual an (n × n)–matrix). Definition 1.1 Let M be an (X × X)matrix. A triple (H, v, w), where H is a Hilbert space and v and w are maps from X to H, is called a Gram representation of M if ∀ x, x ′ ∈ X: Mxx ′ = 〈vx, wx ′ 〉 (1) and if there is a finite constant γM> 0 such that sup x∈X max{‖vx‖, ‖wx‖} ≤ γM. (2) γM is called the Gram constant of M associated to the Gram representation (H, v, w).
A RIGOROUS TREATMENT OF THE PERTURBATION THEORY FOR MANYELECTRON SYSTEMS
, 904
"... Four point correlation functions for many electrons at finite temperature in periodic lattice of dimension d ( ≥ 1) are analyzed by the perturbation theory with respect to the coupling constant. The correlation functions are characterized as a limit of finite dimensional Grassmann integrals. A lower ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Four point correlation functions for many electrons at finite temperature in periodic lattice of dimension d ( ≥ 1) are analyzed by the perturbation theory with respect to the coupling constant. The correlation functions are characterized as a limit of finite dimensional Grassmann integrals. A lower bound on the radius of convergence and an upper bound on the perturbation series are obtained by evaluating the Taylor expansion of logarithm of the finite dimensional Grassmann Gaussian integrals. The perturbation series up to second order is numerically implemented along with the volumeindependent upper bounds on the sum of the higher order terms in 2 dimensional case.
Exponential decay of equaltime fourpoint correlation functions in the Hubbard model on the copperoxide lattice
 Ann. Henri Poincaré, Online First
"... ar ..."
(Show Context)
Axiomatic quantum field theory. Jet formalism
, 707
"... Abstract. Jet formalism provides the adequate mathematical formulation of classical field theory reviewed in hepth/0612182v1. A formulation of QFT compatible with this classical one is discussed. We are based on the fact that an algebra of Euclidean quantum fields is graded commutative, and there a ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Jet formalism provides the adequate mathematical formulation of classical field theory reviewed in hepth/0612182v1. A formulation of QFT compatible with this classical one is discussed. We are based on the fact that an algebra of Euclidean quantum fields is graded commutative, and there are homomorphisms of the graded commutative algebra of classical fields to this algebra. As a result, any variational symmetry of a classical Lagrangian yields the identities which Euclidean Green functions of quantum fields satisfy. Jet manifold formalism provides the adequate mathematical formulation of classical field theory, called axiomatic classical field theory (henceforth ACFT) (see [66] for a survey). Bearing in mind quantization, we consider a graded C ∞ (X)module of even and odd classical fields on a smooth manifold X = R n, n ≥ 2, coordinated by (x λ). In ACFT, these fields are represented by sections s of a graded vector bundle Y = Y0 ⊕ Y1 → X coordinated by (x λ, y a). Finite order jet manifolds J r Y, r = 1,..., of Y → X are also vector bundles over X coordinated by (x λ, y a, y a Λ), Λ  = k ≤ r, where Λ = (λ1,...,Λk) denote symmetric multiindices. Sections of Y and the jet bundles J r Y → X generate a graded commutative C ∞ (X)algebra P 0 of polynomials of (y a, y a Λ). The differential graded algebra